A Primer on the Orthogonal GARCH Model

Transcription

1 1 A Primer on the Orthogonal GARCH Model Professor Carol Alexander ISMA Centre, The Business School for Financial Markets, University of Reading Keywords: Principal component analysis, covariance matrix, orthogonal factor model, Value-at-Risk, generalised autoregressive conditional heteroscedasticity, exponentially weighted moving average. 1. Introduction During the last few years there have been many changes in the way that financial institutions model risk. New risk capital regulations have motivated a need for vertically integrated risk systems based on a unified framework throughout the whole office. If the risk exposures in all locations of a large institution are to be netted, the risk system must also be horizontally integrated and regulators are pushing towards an environment where traders, quants and risk managers from all offices are referring to risk measures generated by the same models. This is a huge task which remains a challenge for many financial institutions, but the result should be worthwhile: a set of simple Value-at-Risk (VaR) measures which can be used to allocate capital between different areas of the firm, and to set traders limits and levels of capital reserves. International financial market regulators require these VaR measures to be produced on a daily basis by all financial institutions, and have imposed stringent requirements for the internal models used to produce these numbers. At the heart of many of the internal VaR models, which are currently, being implemented by financial institutions lies a large covariance matrix. Fully vertically and horizontally integrated systems would use this same matrix to price, hedge and measure risk in a unified framework across the entire firm. Typically hundreds of risk factors would need to be accounted for in a very large dimensional covariance matrix. Because of the complexity of the problem often very simple measures of volatility and correlation are used in this covariance matrix. For example the RiskMetrics methodologies designed by JP Morgan use either equally weighted moving averages or exponentially weighted moving averages with the same smoothing constant for all returns. However both methods have substantial limitations, as described in Alexander (1996).

2 2 The purpose of this paper is to show how orthogonal factor models can be used to simplify the process of producing these large covariance matrices on a daily basis. Its central idea is the orthogonal factorisation by principal component analysis of the assets or risk factors that the covariance matrix represents. The paper focuses on how these principal components can be used in conjunction with standard volatility estimation methods, such as equally weighted or exponentially weighted moving averages or generalised autoregressive conditional heteroscedasticity (GARCH), to produce large positive semi definite covariance matrices. In a highly correlated system only a few principal components are required to represent the system variation to a very high degree of accuracy. For example if 3 principal components were used to represent a term structure with n maturities, only 3 volatilities will need to be calculated following the method described in this paper. This is much more simple than calculating all of the n(n+1)/2 volatilities and correlations directly. The covariance matrix that is constructed using the principal components method is guaranteed to be positive semi-definite. Also by tailoring the number of principal components to capture only the main variations, rather than the small and insignificant movements that might better be ascribed to 'noise', correlation estimates will be more stable than if they were estimated directly. The first part of the paper deals with the orthogonalization of risk factors in a multi-factor model using principal component analysis. Sections 2 and 3 cover the algebra of the method for generating a full nxn covariance matrix, which will always be positive semi-definite, from the volatilities of the principal components alone. The basic algebra is illustrated by an example that uses equally weighted moving average estimates of the volatilities of the principal components. The next section extends the basic model to allow exponentially weighted moving average variances of the principal components. Exponentially weighted moving averages of the squares and cross products of returns are a standard method for generating covariance matrices. But a limitation of this type of direct application of exponentially weighted moving averages is that the covariance matrix is only guaranteed to be positive semi-definite if the same smoothing constant is used for all the data. That is, the reaction of volatility to market events and the persistence in volatility must be assumed to be the same in all the markets that are represented in the covariance matrix. A major advantage of the orthogonal factor method described here is that it allows exponentially weighted moving average methods to be used without this

3 3 unrealistic constraint. In fact the smoothing constant that defines the exponential volatility of any particular asset or market will be given by its factor weights in the principal components representation. Put another way, the volatility persistence and market reaction of a particular returns series will not be the same as that of the other variables in the system, but instead it is related to its correlation with those variables. Having applied the orthogonal method with exponentially weighted moving averages in section 4, it is a small step to replace exponentially weighted moving average variances with fully fledged GARCH variances as described in section 5. The orthogonal GARCH model is validated empirically in section 6 using data on commodity futures and interest rates. In section 7 it is applied to systems of equity indices. A main focus of this part of the paper is on the calibration of volatilities and correlations that are generated using the orthogonal GARCH model with those that are generated directly using standard multivariate GARCH parameterisations (see Engle and Kroner, 1993). These examples have been included to show that the model calibration will need much more care when the system is not so highly correlated. But although the initial calibration of the orthogonal GARCH model may require some care, once the model has been calibrated in this way it may be used on a daily basis without recalibration. Orthogonal GARCH has a number of advantages over direct multivariate GARCH: Since only very few univariate GARCH models are required to generate the large covariance matrix, convergence problems of the optimization routines will be rare (whereas they are common place with the application of direct multivariate GARCH to large systems); There need be no constraints on dimensionality of the original system (whereas direct multivariate GARCH models can only really cope with single figure dimensions); The orthogonal GARCH method gives one the option of cutting out any 'noise' in the data that would otherwise make correlation estimates unstable; And the orthogonal method allows one to generate estimates for volatilities and correlations of variables in the system even when data are sparse and unreliable, for example in illiquid markets. Section 8 shows how the orthogonal factor method can be applied to generate very large covariance matrices. It suggests how one should divide risk factors into different categories before the application of principal components analysis, and demonstrates how best to 'splice' together different blocks into a covariance matrix of the original large system.

4 4 This is a long paper to summarise and conclude. But the main points are quite simple. First, in the world of financial markets where there is so much uncertainty, it makes sense to distil the important information into a few factors that influence all the variables in the system to a greater or lesser extent. So much of market variability can be just put down to 'noise' and models that do not know how to filter that out may lack robustness. And second, the orthogonal method allows one to use GARCH models to generate large covariance matrices for many types of systems, from equities or foreign exchange rates to all types of term structures. This primer begins by showing how the orthogonal model may be applied with exponentially weighted moving average variance, but there are very good reasons to prefer GARCH models to exponentially weighted moving average models. Perhaps the most important reason is the convergence of GARCH volatility and correlation forecasts to their long-term average levels, whereas the exponentially weighted moving average model has a constant term structure of forecasts. The paper has been written in response to the many requests that I have received since first explaining these ideas. First and foremost it is written as a primer on the method, and so I have provided the examples and supporting programs as free downloads from the ISMA centre web site ( Hyperlinks within this paper will take you to the programs, which may be run on TSP ( with the zip file of data provided. A demo version of TSP is available free from their website, but it has very restricted memory capabilities so most of the example programs presented here could not be run on the demo version of TSP. Commercial software for orthogonal GARCH is also available from Algorithmics Inc ( Whilst this paper provides a thorough empirical validation of orthogonal GARCH models for equities, commodities and interest rates it does not attempt to provide any theoretical results on the statistical properties of orthogonal GARCH models. It is hoped that the work presented here will motivate some readers towards more theoretical work on the model. 2. Principal Components Analysis The ideas of this section are illustrated by generating covariance matrices for two sets of returns (a) a term structure of crude oil futures, and (b) a small set of French equities. Each of these systems is more or less correlated: the crude oil futures far more than the equities. If one were to generate a covariance matrix for

5 5 each system by applying some simple weighted average measures of variance and covariance to the returns, it would certainly be found that the crude oil futures correlations are higher and more stable over time than the equity correlations. The crude oil futures are highly collinear because there are only a few important sources of information in the data, which are common to many variables. Principal components analysis (PCA) is a method for extracting the most important independent sources of information in the data. From a set of k stationary returns it will give up to k orthogonal stationary variables which are called the principal components. At the same time PCA states exactly how much of the total variation in the original data is explained by each principal component. The results of PCA are sensitive to re-scaling of the data, and so it is standard practice to normalise the data before the analysis. We therefore assume that each column in the stationary data matrix X has mean zero and variance 1, having previously subtracted the sample mean and divided by the sample standard deviation. Other forms of normalisation are occasionally applied, which explains why statistical packages may give different results. Let the columns of X be x 1,., x k so that X'X is a kxk symmetric matrix with 1's along the diagonal, of correlations between the variables in X. Each principal component is a linear combination of these columns, where the weights are chosen from the set of eigenvectors of the X'X correlation matrix so that (a) the first principal component explains the maximum amount of the total variation in X, the second component explains the maximum amount of the remaining variation, and so on; and (b) the principal components are uncorrelated with each other. Denote by W the matrix of eigenvectors of X'X. Thus X'X W = W Λ where Λ is the diagonal matrix of eigenvalues of X'X. Order the columns of W according to size of corresponding eigenvalue. Thus if W = (w ij ) for i,j = 1,...k, then the mth column of W, denoted w m = (w 1m,.,w km )' is the kx1 eigenvector corresponding to the eigenvalue λ m and the column labelling has been chosen so that λ 1 > λ 2 >... >λ k. Then define the mth principal component of the system by p m = w 1m x 1 + w 2m x w km x k where x i denotes the ith column of X, or in matrix notation

6 6 p m = Xw m Each principal component is a time series of linear combinations of the X variables, and if these are placed as the columns of a full Txk matrix P of principal components we have P = XW (1) To see that this procedure leads to uncorrelated components, note that P'P = W'X'XW = W'WΛ. But W is an orthogonal matrix, that is W' = W -1 and so P'P = Λ. Since this is a diagonal matrix the columns of P are uncorrelated. Since W' = W -1 (1) is equivalent to X = PW', that is X i = w i1 P 1 + w i2 P w ik P k (2) where X i and P i denote the ith columns of X and P respectively. Thus each data vector in X is a linear combination of the principal components. The proportion of the total variation in X that is explained by the mth principal component is λ m /(sum of eigenvalues). The principal components method is first illustrated by a standard example: an analysis of US semiannualised zero coupon rates using monthly data from 1944 to , shown in figure 1. Such a long period of data is very useful to illustrate the stylised facts of PCA, but is of little practical significance. The input from these data to a PCA is the returns correlation matrix X'X (table 1). For the US data this matrix exhibits the typical structure of the yield curve: correlations tend to decrease with the spread, and the 1mth rate and long rate have lower correlations with other rates since they are most influenced by fiscal and monetary policy. 1 Copyright Thomas S. Coleman, Lawrence Fisher, Roger G. Ibbotson, U.S Treasury Yield Curves, 1993 Edition,Ibbotson Associates, Chicago.

7 7 Figure 1: US Zero-Coupon Yields /31/44 2/28/49 3/31/54 4/3/59 5/28/64 6/3/69 7/31/74 8/31/79 9/28/84 1/31/89 1 mth 3 mth 6 mth 9 mth 12 mth 18 mth 2 yr 3 yr 4 yr 5 yr 7 yr 1 yr 15 yr 2 yr LONG Table1: Correlation Matrix for US Zero Coupon Rates 1mth 3mth 6mth 9mth 12mth 18mth 2yr 3yr 4yr 5yr 7yr 1yr 15yr long 1mth 1 3mth mth mth mth mth yr yr yr yr yr yr yr long

8 8 The output from PCA is summarised in table 2: The table 2a gives the eigenvalues and corresponding amount of variation in the original system that is explained, for the first three principal components. Table 2a: Eigenvector Analysis Component Eigenvalue Cumulative R 2 P P P The trace of the X'X matrix above is 15, the sum of the diagonal elements, which is the number of variables in the system. The diagonal matrix of its eigenvalues has the same trace as X'X (because trace is invariant under similarity transforms) and so the sum of the eigenvalues is also 15. The largest eigenvalue is 11.1, so the proportion of total variation that it explains is 11.1/15, or 78.6%. The second largest eigenvalue, 1.632, explains a further 1.632/15 that is 11.7% and the third largest eigenvalue (.4963) explains another 3.5% of the total variation. Thus 93.8% of the total variation in the zero coupon bond returns is explained by the linear model with just 3 principal components. The second part of the output from PCA is a kxk matrix of factor weights, W given in table 2b.Only the factor weights corresponding to the first three principal components are shown below, and they exhibit certain stylised facts. First note that the weights on the first principal component w I1 are similar, except perhaps for the very short and very long maturities that have lower correlation with the rest of the system. But in general the correlations are quite high, and this is reflected in the similarity of the factor weights w I1. Under perfect correlation X'X is simply a matrix of 1's, with rank 1 and the single eigenvector (1,1,.,1)'. For independent systems with full rank, the first eigenvector corresponding to the largest eigenvalue will take values less than 1, but the more highly correlated the variables the larger and more similar are the eigenvector values corresponding to the largest eigenvector. Put another way, the factor weights on the first principal component will be large, and similar for all variables in a highly correlated system. An upwards shift in the first principal component therefore induces a roughly parallel shift in the yield curve, and for this reason the first principal component is called the trend component of the yield curve, and in this example it explains 78.6% of the total variation over the period.

9 9 Table 2b: Factor Weights P1 P2 P3 1mth mth mth mth mth mth yr yr yr yr yr yr yr long The factor weights on the second principal component, w i2, are monotonically decreasing from.5727 on the 1mth rate to on the long rate. Thus an upward movement in the second principal component induces a change in slope of the yield curve, where short maturities move up but long maturities move down. The second principal component is called the tilt and in this example 11.7% of the total variation is attributed to changes in slope. The factor weights on the third principal component, w i3, are positive for the short rates, but decreasing and becoming negative for the medium term rates, and then increasing and becoming positive again for the longer maturities. So the third principal component influences the convexity of the yield curve, and in this example 3.5% of the variation during the data period is due to changes in convexity.

10 1 3. Generating a Covariance Matrix for a Single Risk Factor Category Now consider how principal components may be used to generate a small covariance matrix, such as the covariance matrix for a yield curve, or the covariance matrix for a set of equities, or a set of equity indices. First suppose that the ith asset return is y i, so that the normalised variables are x i = (y i - µ i )/σ i where µ i and σ i are the mean and standard deviation of y i for i = 1, k. Write the principal components representation as y i = µ i + ω* i1 p 1 + ω* i2 p ω* im p m + ε i (3) where ω * ij = w ij σ i and the error term in (3) picks up the approximation from using only m of the k principal components. Since principal components are orthogonal their covariance matrix is diagonal. The variances of the principal components can be quickly transformed into a covariance matrix of the original system using the factor weights: Taking variances of (3) gives V = ADA' + V ε (4) where A = (ω * ij ) is the matrix of normalised factor weights, D = diag(v(p 1 ),... V(P m )) is the diagonal matrix of variances of principal components and V ε is the covariance matrix of the errors. Thus the full kxk covariance matrix of asset returns V is obtained from a just a few estimates of the variances of the principal components, and the covariances of the errors. However V may not be positive definite. 2 Although D is positive definite because it is a diagonal matrix with positive elements, there is nothing to guarantee that ADA' will be positive definite when m < k. To see this write x'ada'x = y'dy where A'x = y. Since y can be zero for some non-zero x, x'ada'x will not be strictly positive for all nonzero x. Of course V would be positive definite if ADA' were positive definite, because V ε is positive

11 11 definite. So if a good approximation has been achieved with m < k principal components there is a reasonable chance that V ε will be strictly positive definite. However if it is only positive semi-definite some weights x could give zero portfolio variance. But when covariance matrices are based on (4) with m < k, they can always be run through an eigenvalue check to ensure strict positive definiteness. The only way to guarantee strict positive definiteness without having to check is to take all k principal components in the factor model, in which case there is no error term and V ε =. Just to illustrate the procedure, consider the returns to three stocks in the CAC 4: Paribas, SocGen and Danone using daily data from 1 st Jan 1994 to 9 th Feb The direct calculation of their covariance matrix, using equally weighted data over the whole period, is Table 3a: Correlation Matrix Paribas SocGen Danone Paribas SocGen Danone The same matrix may be obtained using ADA' where A is the matrix of rescaled principal components factor weights and D is the diagonal matrix of variances of the principal components. To see this, first perform PCA with the full number of components, and this gives the following output: Table 3b: Eigenvalue Analysis Component Eigenvalue Cumulative R 2 P P P A symmetric matrix A is positive definite if x'ax > for all non-zero x. A is positive definite if and only if all its eigenvalues are positive (as can be seen by writing A = C'ΛC where Λ is the diagonal matrix of eigenvalues of A).

12 12 Table 3c: Factor Weights P1 P2 P3 Paribas SocGen Danone The matrix A is obtained by multiplying each factor weight by the corresponding standard deviation: Std Dev Paribas SocGen Danone So the matrix A is: Since in this case we are just taking equally weighted variance estimates over the whole period, and since the data were normalised before the analysis, the principal components all have unit variance. That is, D = I, the 3x3 identity matrix. So ADA' = AA' and the reader may verify that this gives the same covariance matrix as the one calculated directly above. The example above may be reproduced by the reader using the program ex1.tsp. It has been included simply to illustrate the method, but clearly there is nothing to be gained from the method when equally weighted average variances are employed. Equally weighted averages of the squares are unbiased estimates of the unconditional variance 3, and each principal component will have a variance estimate of 1 if the estimate is taken over the same data period as the PCA. But suppose exponentially weighted average estimates of the unconditional variance were employed instead? These have the substantial advantage of

13 13 responding better to current market circumstances, and being less affected by stress events far in the past, than equally weighted averages (see Alexander, 1998). On the other hand they have the disadvantage of there being no one best method for choosing an optimal value of the smoothing constant. 4. Generating Covariance Matrices using Exponentially Weighted Moving Average Variances of the Principal Components. When exponentially weighted moving average (EWMA) volatilities and correlations are estimated directly, the decay factor, as defined by the smoothing constant, must be the same for all series in a large covariance matrix, otherwise it may not be positive semi-definite (see the RiskMetrics Technical Document on But when exponentially weighted moving average volatilities and correlations are estimated indirectly using the orthogonal factor method just described, each volatility and correlation will have a different decay factor. Even if the EWMA variances of the principal components all had the same smoothing constant 4 the transformation of these variances using factor weights, by equation (4), will induce different decay rates for the variances and covariances of the variables in the original system. So, provided all principal components are retained, the method provides a simple way to apply EWMA to generate a large positive definite covariance matrix. The program ex2.tsp uses the same French equity data as ex1.tsp but with exponentially weighted moving averages. Figure 2 plots the volatilities and correlations that are obtained using the orthogonal method with the volatilities and correlations that are obtained using exponentially weighted moving averages directly on the squared returns. This is a very basic example, so the smoothing constant has simply been set as.95 for all exponentially weighted moving averages (later examples will use different values of the smoothing constant for the principal components). 3 One does not normally take squared mean deviations with a bias correction (n-1) in the denominator, since it makes no discernable difference for variance estiamtors based on daily financial returns. 4 Choosing identical smoothing constants for all principal components is in fact neither necessary for positive definiteness nor desireable for optimal forecasting. The optimal smoothing constants may be lower for the higher, less important principal components, wheras the volatility of the first, trend component may be the most persistent of the principal component volatilities in a highly correlated system.

14 14 Figure 2: Comparison of Direct and Orthogonal EWMA Volatilities and Correlations Jun-94 Oct-94 Feb-95 Jun-95 Oct-95 Paribas Direct EWMA Volatility Feb-97 Jun-97 Oct-97 Feb-98 Jun-98 Oct-98 Paribas Orthogonal EWMA Volatility Jun-94 Oct-94 Feb-95 Jun-95 Oct-95 Feb-97 Jun-97 Oct-97 Feb-98 Jun-98 Oct-98 Paribas & Soc Gen Direct EWMA Correlation Paribas & Soc Gen Orthogonal EWMA Correlation Jun-94 Oct-94 Feb-95 Jun-95 Oct-95 Soc Gen Direct EWMA Volatility Feb-97 Jun-97 Oct-97 Feb-98 Jun-98 Oct-98 Soc Gen Orthogonal EWMA Volatility Jun-94 Oct-94 Feb-95 Jun-95 Oct-95 Feb-97 Jun-97 Oct-97 Feb-98 Jun-98 Oct-98 Paribas & Danone Direct EWMA Correlation Paribas & Danone Orthogonal EWMA Correlation Jun-94 Oct-94 Feb-95 Jun-95 Oct-95 Danone Direct EWMA Volatility Feb-97 Jun-97 Oct-97 Feb-98 Jun-98 Oct-98 Danone Orthogonal EWMA Volatility Jun-94 Oct-94 Feb-95 Jun-95 Oct-95 Feb-97 Jun-97 Oct-97 Feb-98 Jun-98 Oct-98 Danone & Soc Gen Direct EWMA Correlation Danone & Soc Gen Orthogonal EWMA Correlation

15 15 A comparison of these plots is a crucial part of the orthogonal model calibration. If these volatilities and correlations are not similar it will be because (a) the data period used for the PCA is too long, or (b) there are variables that are included in the system that are distorting the volatilities and correlations of other variables computed using the orthogonal method. Both these problems arise if there is insufficient correlation in the system for the method to be properly applied. If one or more of the variables have a low degree of correlation with the other variables over the data period, the factor weights in the PCA will lack robustness over time. The model could be improved by using a shorter data period, and/or omitting the less correlated variables from the system. Having detailed the method, let us now see its real strength by applying it to a larger and highly correlated system. The program ex3.tsp applies the orthogonal method using just 3 principal components to the WTI crude oil futures data on all monthly maturities from 1 month to 12 months, sampled daily between 4 th February 1993 and 24 th March The 1, 2, 3, 6, 9 and 12-month maturity futures prices are shown in figure 3, and see Alexander (1999) for a full discussion of these data and of correlations in energy markets in general. 5 Figure 3: NYMEX Sweet Crude Prices m1 m2 m3 m6 m9 m12 5 Many thanks to Enron for providing these data.

16 16 This very highly correlated system is ideally suited to the use of PCA. The output from PCA given in table 4 below shows that 99.8% of the variation in the system may be explained by just 3 principal components. In fact just the first principal component explains almost 96% of the variation over the period, and with two principal components over 99% of the variation is explained. Of course the factor weights show that, as with any term structure, the interpretations of the first three principal components are the trend, tilt and curvature components respectively. Table 4a: Eigenvalue Analysis Component Eigenvalue Cumulative R 2 P P P Table 4b: Factor Weights P1 P2 P3 1mth mth mth mth mth mth mth mth mth mth mth mth The great advantage in using the orthogonal method on term structure data is that all the volatilities and correlations in the system can be derived from just 3 exponentially weighted moving average variances.

17 17 That is, instead of estimating 78 exponentially weighted moving average volatilities and correlations directly, using the same value of the smoothing constant throughout, only 3 exponentially weighted moving average variances of the trend, tilt and curvature principal components need to be generated. In some term structures, including the crude oil futures term structure used in example 3, only 2 components already explain over 99% of the variation, so adding a 3 rd component makes no discernible difference to the covariance results. Figure 4: Comparison of Direct and Orthogonal EWMA Volatilities Feb-93 Oct-93 Jun-94 Oct-94 Feb-95 Jun-95 Oct-95 Feb-97 Jun-97 Oct-97 Feb-98 Jun-98 Oct-98 Feb-99 2mth Direct EWMA Volatility 2mth Orthogonal EWMA Volatility Feb-93 Oct-93 Jun-94 Oct-94 Feb-95 Jun-95 Oct-95 Feb-97 Jun-97 Oct-97 Feb-98 Jun-98 Oct-98 Feb-99 6mth Direct EWMA Volatility 6mth Orthogonal EWMA Volatility Feb-93 Oct-93 Jun-94 Oct-94 Feb-95 Jun-95 Oct-95 Feb-97 Jun-97 Oct-97 Feb-98 Jun-98 Oct-98 Feb-99 12mth Direct EWMA Volatility 12mth Orthogonal EWMA Volatility

18 18 All the volatilities and correlation variances of the original system can be recovered using simple transformations of the diagonal matrix of principal component variances, as is done in ex3.tsp. Moreover the principal component variances may use different smoothing constants. In example 3 the default value of.95 for the 1 st, 2 nd and 3 rd principal components has been used, but the reader may be interested to experiment with using different smoothing constants for the principal component variances. Even if the principal component variances do all have the same smoothing constant, the volatilities of different maturities in the term structure would have different exponential smoothing properties. This is of course because they have different factor weights in the principal component representation. The figures in figure 4 show some of the volatilities that are generated using the orthogonal method compared with directly estimated exponentially weighted moving average volatilities. The coincidence of the results obtained by the orthogonal method with those obtained by direct estimation shows how powerful this method is. From just 2 or 3 exponentially weighted moving averages, the entire 12x12 covariance matrix of the original system is recovered with negligible loss of precision. Figure 5 shows some of the correlations obtained using the orthogonal method for different pairs of maturities. The reader may easily view more of them from the off-diagonal elements of the covariance matrix in ex3.tsp. The orthogonal EWMA correlations shown in figure 5 are very similar indeed to the correlations generated by direct EWMA. But there is a problem with using exponentially weighted averages at all in the crude oil futures market. Although they contain fewer 'ghost features' and other artificial effects that result from the use of equally weighted moving averages, there is still a disturbing lack of correlation between some of the near maturity futures. This problem will be a point of discussion based on figure 8 below, where the same correlations are measured by the orthogonal GARCH model. Figure 5: Some of the Correlations from the Orthogonal EWMA Model Feb-93 Oct-93 Jun-94 Oct-94 Feb-95 Jun-95 Oct-95 Feb-97 Jun-97 Oct-97 Feb-98 Jun-98 Oct-98 Feb-99 1mth-3mth 3mth-6mth 6mth-9mth 9mth-12mth

19 19 There are many advantages with the orthogonal method for generating covariance matrices, even when it is applied using only exponentially weighted moving average variance estimates. Obviously the computational burden is much lighter when all of the k(k+1)/2 volatilities and correlations are simple matrix transformations of just 2 or 3 exponentially weighted moving average variances. But also data may be difficult to obtain directly, particularly on some financial assets that are not heavily traded. When data are sparse or unreliable on some of the variables in the system a direct estimation of volatilities and correlation may be very difficult. But if there is sufficient information to infer their factor weights in the principal component representation, their volatilities and correlations may be obtained using the orthogonal method. For example, some bonds or futures may be relatively illiquid for certain maturities, and statistical forecasts of their volatilities may be difficult to generate directly on a daily basis. But the variances of the principal components of the entire term structure can be transformed using the factor weights into a full covariance matrix that generates flexible forecasts of all maturities, including the illiquid ones. 5. Introducing the Orthogonal GARCH Model The univariate generalised autoregressive conditional heteroscedasticity (GARCH) models that were introduced by Engle (1982) and Bollerslev (1986) have been very successful for volatility estimation and forecasting in financial markets. The mathematical foundation of GARCH models compares favourably with some of the alternatives used by financial practitioners, and this mathematical coherency makes GARCH models easy to adapt to new financial applications. There is also evidence that GARCH models generate more realistic long-term forecasts, since the volatility and correlation term structure forecasts will converge to the long-term average level (see Alexander, 1998, 2). As for short-term volatility forecasts, statistical results are mixed (see for example Brailsford and Faff, 1996, Dimson and Marsh, 199, Figlewski, 1994, Alexander and Leigh (1997)). This is not surprising since the whole area of statistical evaluation of volatility forecasts is fraught with difficulty (see Alexander, 2). Another test of a volatility forecasting model is in its hedging performance: and there is much to be said for using the GARCH volatility framework for pricing and hedging options (see Duan 1995, 1996). Engle and Rosenberg (1995) provide an operational evaluation of GARCH models in option pricing and hedging, where their superiority to the Black-Scholes methods stems from the fact that stochastic volatility is already built into the model, so there is no need for additional vega hedging.

20 2 Large covariance matrices that are based on GARCH models would, therefore, have clear advantages. But previous research in this area has met with rather limited success. It is straightforward to generalise the univariate GARCH models to multivariate parameterisations, as in Engle and Kroner (1993). But the actual implementation of these models is extremely difficult. With so many parameters, the likelihood function becomes very flat, and so convergence problems are very common in the optimization routine. If the modeller needs to 'nurse' the model for systems with only a few variables, there is little hope of a fully functional implementation of a direct multivariate GARCH model to work on large risk systems. The idea of using factor models with GARCH is not new. Engle, Ng and Rothschild (199) use the capital asset pricing model to show how the volatilities and correlations between individual equities can be generated from the univariate GARCH variance of the market risk factor. Their results have a straightforward extension to multi-factor models, but unless the factors are orthogonal a multi-variate GARCH model will be required, with all the associated problems. A principal components representation is a multi-factor model. In fact the orthogonal GARCH model is a generalisation of the factor GARCH model introduced by Engle, Ng and Rothschild (199) to a multifactor model with orthogonal factors. The orthogonal GARCH model allows kxk GARCH covariance matrices to be generated from just m univariate GARCH models. Here k is the number of variables in the system and m is the number of principal components used the represent the system. It may be that m can be much less than k, and quite often one would wish m to be less than k so that extraneous 'noise' is excluded from the data. But since only univariate GARCH models are used it does not really matter: there no dimensional restrictions as there are with the direct parameterisations of multivariate GARCH. Of course, the principal components are only unconditionally uncorrelated, so a GARCH covariance matrix of principal components is not necessarily diagonal. However the assumption of zero conditional correlations has to be made, otherwise it misses the whole point of the model, which is to generate large GARCH covariance matrices from GARCH volatilities alone.

21 21 6. Empirical Validation of the Orthogonal GARCH Model using Term Structure Data Before presenting some empirical examples on orthogonal GARCH let us just rephrase the results of section 3 in the framework of stochastic volatility. Thus the mxm diagonal matrix of variances of the principal components is a time-varying matrix denoted D t and the time-varying covariance matrix V t of the original system is approximated by V t = A D t A' (5) where A is the kxm matrix of re-scaled factor weights. The representation (5) will give a positive semidefinite matrix at every point in time, even when the number m of principal components is much less that the number k of variables in the system. However the accuracy of the representation (5) depends on the number of principal components used being sufficient to explain a large part of the variation in the system. The method will therefore work well when principal component analysis works well, i.e. on term structures and other highly correlated systems. The model (5) is called orthogonal GARCH when the diagonal matrix D t of variances of principal components is estimated using a GARCH model. In the examples given here the standard 'vanilla' GARCH(1,1) model is used. The conditional variance at time t is defined as: σ 2 t 2 2 = ω + α ε t 1 + βσ t 1 (6) where the 'market reaction' parameter α and the 'volatility persistence' parameter β should sum to less than one (for convergence of term structure volatility forecasts). In the exponentially weighted moving average model these parameters always sum to one, so the volatility term structure will be constant. The orthogonal GARCH model is particularly useful for term structures where the more illiquid maturities can preclude the direction estimation of GARCH volatilities. When market trading is rather thin there may be little autoregressive conditional heteroscedasticity in the data, and what is there may be rather unreliable. The orthogonal GARCH model has the advantage that the volatilities of such assets, and their correlations with other assets in the system, are derived from the principal component volatilities that are common to all assets and the factor weights that are specific to that particular asset. The first example of the orthogonal GARCH model is a straightforward extension of example 3, using GARCH(1,1) variances of principal components in place of exponentially weighted moving averages. The

22 22 ex4.tsp uses the same crude oil futures term structure data as example 3 and table 5 reports the estimated coefficients in GARCH(1,1) models of the first 2 principal components. Table 5: GARCH(1,1) Models of the First and Second Principal Component 1 st Principal Component 2 nd Principal Component Coefficient t-stat Coefficient t-stat constant.65847e e ω e α β Figure 6: Comparison of Direct and Orthogonal GARCH Volatilities Feb-93 Oct-93 Jun-94 Oct-94 Feb-95 Jun-95 Oct-95 Feb-97 Jun-97 Oct-97 Feb-98 Jun-98 Oct-98 Feb-99 2mth Direct GARCH Volatility 2mth Orthogonal GARCH Volatility Feb-93 Oct-93 Jun-94 Oct-94 Feb-95 Jun-95 Oct-95 Feb-97 Jun-97 Oct-97 Feb-98 Jun-98 Oct-98 Feb-99 6mth Direct GARCH Volatility 6mth Orthogonal GARCH Volatility Feb-93 Oct-93 Jun-94 Oct-94 Feb-95 Jun-95 Oct-95 Feb-97 Jun-97 Oct-97 Feb-98 Jun-98 Oct-98 Feb-99 12mth Direct GARCH Volatility 12mth Orthogonal GARCH Volatility

23 23 Figure 6 is similar to figure 4, the only difference being that GARCH(1,1) models have been used to generate figure 6 wherever exponentially weighted moving averages were used for figures 4. The orthogonal and direct volatilities that are compared in figure 6 are very close indeed. In fact, they are almost identical to the EWMA volatilities illustrated in figure 4. Why bother with GARCH then? There are two important reasons. The first is that EWMA volatility term structure forecasts do not converge to the long-term average, but GARCH forecasts do, provided α + β < 1. In fact the orthogonal GARCH model can be extended quite easily to provide forecasts of the average volatility over the next n days, for any n (see Alexander 1998, 2). The ex5.tsp does precisely this for the crude oil term structure data, producing series of term structure volatility forecasts that converge to a long-term average. Volatility terms structures for the 1mth future are shown in figure 7. 6 Figure 7: Orthogonal GARCH Term Structure Volatility Forecasts for 1mth Crude Oil Futures Feb-93 Oct-93 Jun-94 Oct-94 Feb-95 Jun-95 Oct-95 Feb-97 Jun-97 Oct-97 Feb-98 Jun-98 Oct-98 Feb-99 1-day 5-day 1-day 2-day 3-day 6-day 12-day The second good reason to use orthogonal GARCH rather than orthogonal EWMA is that the orthogonal GARCH correlations will more realistically reflect what is happening in the market. As already mentioned, the correlations shown in figure 5 that were generated by the orthogonal EWMA are a little worrying. One would expect correlations between commodity futures to be more or less perfect most of the time, but the EWMA correlations between the 1 mth futures and other futures, and between other pairs at short maturities, can be considerably below 1 for long periods of time. For example during long periods of 1996 and 1998 the EWMA correlations are nearer to.8 than 1.

24 24 The reason for this is the smoothing constant of.95, which is an appropriate choice for the volatilities (as we know from the comparison of figure 4 with the optimised GARCH models in figure 6) but is clearly too large for the correlations. Unfortunately if one were to reduce the values of the smoothing constants used in the orthogonal EWMA model, so that the correlations were less persistent, so also would the volatilities be less persistent. One can only guess by trial and error what are the approriate values for the smoothing constants, and it may be that there is no clear answer to this question. In the crude oil futures market price decoupling only occurs over very short time spans so correlations may deviate below 1, but only for a short time. Now, if the orthogonal model were to be used with just one principal component (which is possible since the results from section 4 indicate that this trend component explains over 95% of the variation) the correlations would of course be unity. So all the variation in the orthogonal GARCH correlations is coming from the movements in the second principal component. This second principal component is the tilt component, and it explains about 4% of the movement (see section 4). The GARCH(1,1) models of the first two principal components of this term structure, given in the table above, indicate that the second principal component has a lot of reaction (α is about.22) but little persistence (β is about.66). In other words these tilt movements in the term structure of futures prices are intense but short-lived. So one would expect the correlations given by the orthogonal GARCH model in figure 8 to be more accurately reflecting real market conditions than the orthogonal EWMA correlations in figure 5. Figure 8: Some of the Correlations from the Orthogonal GARCH Model Feb-93 Oct-93 Jun-94 Oct-94 Feb-95 Jun-95 Oct-95 Feb-97 Jun-97 Oct-97 Feb-98 Jun-98 Oct-98 Feb-99 1mth-3mth 3mth-6mth 6mth-9mth 9mth-12mth

25 25 The summarise the results so far, this paper has shown how 78 different volatilities and correlations of the term structure of crude oil futures between 1 mth and 12mths can be generated, very simply and very accurately, from just two univariate GARCH models of the first two principal components. It has also shown how volatility forecasts of different maturities can also be generated as simple transformations of these two basic GARCH variances. Now let us now step up a little with the complexity of the data. Still a term structure, but rather a difficult one. The program given in ex6.tsp has been trained on daily zero coupon yield data in the UK with 11 different maturities between 1mth and 1 years from 1 st Jan 1992 to 24 th Mar 1995, shown in figure 9. Figure 9: UK Zero-Coupon Yields Jan-92 Mar-92 May-92 Jul-92 Sep-92 Nov-92 Jan-93 Mar-93 May-93 Jul-93 Sep-93 Nov-93 Jan-94 Mar-94 May-94 Jul-94 Sep-94 Nov-94 Jan-95 Mar-95 1M 2M 3M 6M 12M 2Y 3Y 4Y 5Y 7Y 1Y It is not an easy task to estimate univariate GARCH models on these data directly because yields may remain relatively fixed for a number of days. Particularly on the more illiquid maturities, there may be insufficient conditional heteroscedasticity for GARCH models to converge well. The reader that uses ex6.tsp will see how problematic is the direct estimation of GARCH models on these data. So the orthogonal GARCH volatilities in figure 1 have been compared instead with exponentially weighted moving average volatilities (with a smoothing constant of.9). The orthogonal GARCH volatilities are not as closely aligned with the exponentially weighted moving average volatilities as they were in the previous example, but there is sufficient agreement between them to place a fairly high degree of confidence in the orthogonal GARCH model. Again two principal components were used in the orthogonal GARCH, but the principal component analysis below shows that these two components only account for 72% of the total variation (as opposed to over 99% in the crude oil term structure).

26 26 Table 6a: Eigenvalue Analysis Component Eigenvalue Cumulative R 2 P P P Table 6b: Factor Weights P1 P2 P3 1mth mth mth mth mth yr yr yr yr yr yr Clearly the lower degree of accuracy from a 2 principal component representation is one reason for the observed differences between the orthogonal GARCH volatilities and the EWMA volatilities. Another is that the 1yr yield has a very low correlation with the rest of the system, as reflected by its factor weight on the 1 st principal component, which is quite out of line with the rest of the factor weights on this component. The fit of the orthogonal model is good, but could be improved further if the 1yr bond were excluded from the system.

27 27 The GARCH(1,1) model estimates for the first two principal components are given in table 7 below. This time the second principal component has a better-conditioned GARCH model. So the tilts in the UK yield curve are less temporary and more important than they are in the crude oil term structure discussed above. One consequence of this is the orthogonal GARCH correlations will be less jumpy and more stable than the correlations in figure 8. Table 7: GARCH(1,1) Models of the First and Second Principal Component 1 st Principal Component 2 nd Principal Component Coefficient t-stat Coefficient t-stat constant e ω α β

28 Jan-92 Mar-92 May-92 Jul-92 Sep-92 Nov-92 Jan-93 Mar-93 May-93 Jul-93 Sep-93 Nov-93 Jan-94 Mar-94 May-94 Jul-94 Sep-94 Nov-94 Jan-95 Mar Jan-92 Mar-92 May-92 Jul-92 Sep-92 Nov-92 Jan-93 Mar-93 May-93 Jul-93 Sep-93 Nov-93 Jan-94 Mar-94 May-94 Jul-94 Sep-94 Nov-94 Jan-95 Mar-95 3mth Direct EWMA Volatility 3mth Orthogonal GARCH Volatility 2yr Direct EWMA Volatility 2yr Orthogonal GARCH Volatility Jan-92 Mar-92 May-92 Jul-92 Sep-92 Nov-92 Jan-93 Mar-93 May-93 Jul-93 Sep-93 Nov-93 Jan-94 Mar-94 May-94 Jul-94 Sep-94 Nov-94 Jan-95 Mar Jan-92 Mar-92 May-92 Jul-92 Sep-92 Nov-92 Jan-93 Mar-93 May-93 Jul-93 Sep-93 Nov-93 Jan-94 Mar-94 May-94 Jul-94 Sep-94 Nov-94 Jan-95 Mar-95 6mth Direct EWMA Volatility 6mth Orthogonal GARCH Volatility 4yr Direct EWMA Volatility 4yr Orthogonal GARCH Volatility Jan-92 Mar-92 May-92 Jul-92 Sep-92 Nov-92 Jan-93 Mar-93 May-93 Jul-93 Sep-93 Nov-93 Jan-94 Mar-94 May-94 Jul-94 Sep-94 Nov-94 Jan-95 Mar Jan-92 Mar-92 May-92 Jul-92 Sep-92 Nov-92 Jan-93 Mar-93 May-93 Jul-93 Sep-93 Nov-93 Jan-94 Mar-94 May-94 Jul-94 Sep-94 Nov-94 Jan-95 Mar-95 12mth Direct EWMA Volatility 12mth Orthogonal GARCH Volatility 7yr Direct EWMA Volatility 7yr Orthogonal GARCH Volatility Figure 1: Comparison of Direct EWMA with Orthogonal GARCH Volatilities